Partial Differential Equations and Fluid Dynamics Meeting

Monday 28th October 2019, at the Mathematical Institute: An afternoon of talks by Edriss Titi, Anne-Laure Dalibard, Manuel del Pino and Tim Palmer on the occasion of the retirement of Gregory Seregin.  

Meeting poster:  PDF icon File

Schedule

Lecture room 6

2.00pm Opening remarks

2.05pm Edriss Titi, University of Cambridge

2.50pm Anne-Laure Dalibard, Sorbonne Universite

North atrium

3.35pm: Tea/Coffee

Lecture room 4

4.00pm Manuel del Pino, University of Bath

4.45pm Tim Palmer, University of Oxford

Titles and abstracts

Edriss Titi, University of Cambridge: Mathematical Analysis of Atmospheric Models with Moisture

In this talk I will present some recent results concerning global regularity of certain geophysical models. This will include the three-dimensional primitive equations with various anisotropic viscosity and turbulence mixing diffusion, and certain tropical atmospheric models with moisture. In particular, we will also show the global regularity of the coupled three-dimensional primitive equations with phase change moisture model.  Moreover, in the non-viscous (inviscid) case it can be shown that there is a one-parameter family of initial data for which the corresponding smooth solutions of the primitive equations develop finite-time singularities (blowup).

Capitalizing on the above results, we can provide rigorous justification of the derivation of the Primitive Equations of planetary scale oceanic dynamics from the three-dimensional Navier-Stokes equations, for vanishing small values of the aspect ratio of the depth to horizontal width. Specifically, we can  show that the Navier-Stokes equations, after being scaled appropriately by the small aspect ratio parameter of the physical domain, converge strongly to the primitive equations, globally and uniformly in time, and that the convergence rate is of the same order as the aspect ratio parameter.

Furthermore, we will also consider the singular limit behavior of a tropical atmospheric model with moisture, as $\varepsilon \to 0$, where $\varepsilon >0$ is a moisture phase transition small convective adjustment relaxation time parameter.

Anne-Laure Dalibard, Sorbonne Universite To follow.

Manuel del Pino, University of Bath: Gluing methods for Vortex dynamics in Euler flows 

We consider the two-dimensional Euler flow for an incompressible fluid confined to a smooth domain. We construct smooth solutions with concentrated vorticities around points which evolve according to the Hamiltonian system for the Kirkhoff-Routh energy. The profile around each point resembles a scaled finite mass solution of Liouville's equation. We discuss extensions of this analysis to the case of vortex filaments in 3-dimensional space, along the lines of Da Rios 1904  vortex filament conjecture in connection with the binormal flow of curves. 

 

Tim Palmer, University of Oxford: The Real Butterfly Effect

In this talk I will discuss a paper (*) which I co-authored with Gregory discussing the status of what we call the “real" butterfly effect: the finite-time loss of predictability of the initial value problem based on the Navier-Stokes equations. 

(*) Palmer, T.N., A. Döring and G. Seregin. (2014). The Real Butterfly Effect. Nonlinearity. R123-R141.